Partial regularity of minimizers of p ( x )‐growth functionals with 1 < p ( x ) < 2

We prove partial regularity of minimizers u for p ( x ) ‐energy functionals of the following type: E ( u ) = ∫ Ω( A i j α β( x , u ) D α u i D β u j ) p ( x ) / 2d x , assuming thatA i j α β( x , u )and p ( x ) are sufficiently smooth and that p ( x ) is subquadratic. We prove that u ∈ C 0 , α( Ω 0 )for some α ∈ ( 0 , 1 ) and an open setΩ 0 ⊂ Ω withH m - γ 1( Ω - Ω 0 ) = 0 , whereH s denotes the s ‐dimensional Hausdorff measure andγ 1 = inf Ω p ( x ) .